Recording soft body deformations during simulation in bge
You can write a BGE script that reads the vertex positions from the mesh. 

Store this positions in a way that you can match them with the vertices of the according Blender object.

When you run the BGE embedded you can access Blender (bpy) while your game session is running. Be aware changes will apply AFTER the game session ended.

Therefore I suggest:


read the BGE mesh vertex 
write to Blender object data (e.g. create shape keys)


I do not have a ready to use script nor do I want to write one here. This answer means is more an idea than a complete solution.